Optical properties of photon echoes stimulated by three frequencies

A gaseous atomic or molecular system with four energy levels Eₐ<Eb<Ec<Ed which is stimulated by resonant laser pulses at frequencies ω₁ωba, ω₂ωcb, ω₃ωdc is considered. The six photon echoes which are stimulated by first and second pulses at frequencies ω₁, ω₂, ω₃ are discussed. For wave fronts which are described by e^i(φ-ωt) the echo phase is described by φE=Σ(2φₘII-φₘI) at the sum frequency ω₁+ω₂+ω₃ with $m=1,2,3$, at ω₁+ω₂ with $m=1,2$, at ω₁ with $m=1$, etc. For waves in the z direction where φ=κ(x,y,z)+kz the gradient of the phase yields the wave-vector direction ${{{→}}{k}}E={Σ}(2{{{→}}{k}}ₘII{-}{{{→}}{k}}ₘI)$ in a physically thin sample. A probe pulse at ${{ω}}₂$ and ${{ω}}₃$ after the second pulse can generate a probe-pulse echo at ${{ω}}₁$ with phase ${{φ}}₁={Σ}(2{{φ}}ₘII{-}{{φ}}ₘI){-}{{φ}}₂S{-}{{φ}}₃S$, etc., for other frequency arrangements. Transverse phases ${{κ}}ₘ(xy)$ can be added and subtracted, but no application is suggested for this novel feature. When all ${{κ}}ₘ$ except one are zero, phase conjugation or phase-adaptive optics can occur.